Are there many examples of smooth closed convex surfaces defined in a implicit way?
The only example that comes to my mind are the ellipses $f^{-1}(0)$ where
$$ f(x,y,z)=x^2/a^2+y^2/b^2+z^2/c^2-1, \quad x,y,z\in\Bbb R. $$
Closed means that it is compact and without boundary and convex means that for each point of the surface, it is in one of the half spaces determined by its tangent plane.